On the Approximability of the Yolk in the Spatial Model of Voting

In the spatial model of voting, the yolk and LP (linear programming) yolk are important solution concepts for predicting outcomes for a committee of voters. McKelvey and Tovey showed that the LP yolk provides a lower bound approximation for the size of the yolk and there has been considerable debate on whether the LP yolk is a good approximation of the yolk. In this paper, we show that for an odd number of voters in a two-dimensional space that the yolk radius is at most twice the size of the LP yolk radius. However, we also show that (1) even in this setting, the LP yolk center can be arbitrarily far away from the yolk center (relative to the radius of the yolk) and (2) for all other settings (an even number of voters or in dimension $k\geq 3$) that the LP yolk can be arbitrarily small relative to the yolk. Thus, in general, the LP yolk can be an arbitrarily poor approximation of the yolk.